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C$_3$N$_4$ is a recently discovered phase of carbon nitrides with thetetragonal crystal structure [D.Laniel $\textit{et al.}$, Adv. Mater. 2023,2308030] that is stable at ambient conditions. C$_3$N$_4$ is a semiconductorexhibiting flat-band anomalies in the valence band, suggesting the emergence ofmany-body instabilities upon hole doping. Here, using state-of-the-artfirst-principles calculations we show that hole-doped C$_3$N$_4$ reveals strongelectron-phonon coupling, leading to the formation of a gapped superconductingstate. The phase transition temperatures turn out to be strongly dependent onthe hole concentration. We propose that holes could be injected into C$_3$N$_4$via boron doping which induces, according to our results, a rigid shift of theFermi energy without significant modification of the electronic structure.Based on the electron-phonon coupling and Coulomb pseudopotential calculatedfrom first principles, we conclude that the boron concentration of 6 atoms pernm$^3$ would be required to reach the critical temperature of $\sim$36 K atambient pressure.

Strong electron-phonon coupling and phonon-induced superconductivity in tetragonal C$_3$N$_4$ with hole doping